Abstract
In this chapter we discuss two fundamental problems of discrete geometry, the best-packing problem and the best-covering problem. In Section 3.1 the general best-packing problem is introduced. We show that it is the limiting case as \(s\rightarrow \infty \) of the Riesz s-energy problem, see Proposition 3.1.2. In that section we also estimate the minimal pairwise separation of an N-point s-energy minimizing configuration on a path connected compact set. Section 3.2 introduces the general best-covering problem and discusses the basic relationship between best-packing distance and mesh ratio on a given compact set.
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Notes
- 1.
See (2.7.1) for the definition.
- 2.
A non-empty subset B of a metric space A is called path connected if for any two points \(x, y\in B\), there is a continuous function (“path”) \(\gamma :[0,1]\rightarrow A\) such that \(\gamma (0)=x\), \(\gamma (1)=y\), and \(\gamma ([0,1])\subset B\).
- 3.
Some authors define this ratio as \(2\eta (\omega _N, A)/\delta ^\rho (\omega _N)\).
- 4.
The great circle equidistant from the antipodal points
- 5.
Recall that, as introduced in Section 1.3, \(\mathcal {H}_2\) denotes 2-dimensional Hausdorff measure.
- 6.
The definition of the \(E_8\) lattice is given in Subsection 5.8.3.
- 7.
The definition of the Leech lattice is given in Subsection 5.8.4.
- 8.
In fact, we have \(U_{a,q}=q\mathbb Z^p \cup D_{a, q}\); however, for our proof we only need a one-sided inclusion.
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© 2019 Springer Science+Business Media, LLC, part of Springer Nature
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Borodachov, S.V., Hardin, D.P., Saff, E.B. (2019). Introduction to Best-Packing and Best-Covering. In: Discrete Energy on Rectifiable Sets. Springer Monographs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-84808-2_3
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DOI: https://doi.org/10.1007/978-0-387-84808-2_3
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